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Three scientists named Hutchinson, Rice and Rosengren independently evaluated the character of crack-tip stress fields in the case of power-law-hardening materials. Hutchinson evaluated both plane stress and plane strain, while Rice and Rosengren considered only plane-strain conditions. Both articles, which were published in the same issue of the Journal of the Mechanics and Physics of Solids, argued that stress times strain varies as 1/r near the crack tip, although only Hutchinson was able to provide a mathematical proof of this relationship. Hutchinson began by defining a stress function Φ for the problem. The governing differential equation for deformation plasticity theory for a plane problem in a Ramberg-Osgood material is more complicated than the linear elastic case:
∆^(4)Φ + γ(Φ,σ,r,n,α) = 0
where the function γ differs for plane stress and plane strain. For the Mode I crack problem, Hutchinson chose to represent Φ in terms of an asymptotic expansion in the following form:
Φ = A(θ)r^(s) + B(θ)r^(t) + .......
where A,B are constants that depend on θ, the angle from the crack plane. Equation (A2) is analogous to the Williams expansion for the linear elastic case. If s < t, and t is less than all subsequent exponents on r, then the first term dominates as r →0. If the analysis is restricted to the region near the crack tip, then the stress function can be expressed as follows:
Φ=k σ_0 r^s Φ(θ)
where k is the amplitude of the stress function and is a dimensionless function of Φ. Although Equation (A1) is different from the linear elastic case, the stresses can still be derived. Thus the stresses, in polar coordinates, are given by
σ_rr = K σ_0 r^(s-2) (σ_rr ) ̌(θ) = K σ_0 r^(s-2)(s Φ ̌+Φ ̌")
σ_θθ = K σ_0 r^(s-2) (σ_θθ ) ̌(θ) = K σ_0 r^(s-2) s (s-1) ̌
σ_rθ = K σ_0 r^(s-2) (σ_rθ ) ̌(θ) = K σ_0 r^(s-2)(1-s) Φ ̌'
σ_e = K σ_0 r^(s-2) (σ_e ) ̌(θ) = K σ_0 r^(s-2) 〖(〖 σ_rr〗^2+〖σ_θθ〗^2-σ_rr σ_θθ +3〖σ_rθ〗^2)〗^(1/2)
The boundary conditions for the crack problem are as follows:
Φ(±π) = Φ'(±π) = 0
In the region close to the crack tip where Equation (A3) applies, elastic strains are negligible compared to plastic strains; only the second term in Equation (A1) is relevant in this case. Hutchinson substituted the boundary conditions and Equation (A3) into Equation (A1) and obtained a nonlinear eigenvalue equation for s. He then solved this equation numerically for a range of n values. The numerical analysis indicated that s could be described quite accurately (for both plane stress and plane strain) by a simple formula:
s = (2n+1)/(n+1)
which implies that the strain energy density varies as 1/r near the crack tip. This numerical analysis also yielded relative values for the angular functions σ_ij. The amplitude, however, cannot be obtained without connecting the near-tip analysis with the remote boundary conditions. The J contour integral provides a simple means for making this connection in the case of small-scale yielding. Moreover, by invoking the path-independent property of J, Hutchinson was able to obtain a direct proof of the validity of Equation (A5).